Amy M. MarconnetDepartment of Mechanical Engineering,

Massachusetts Institute of Technology,

Cambridge, MA 01239

e-mail: amymarco@mit.edu

Mehdi Asheghi

Kenneth E. GoodsonFellow ASME

Department of Mechanical Engineering,

Stanford University,

Stanford, CA 94305

From the Casimir Limit toPhononic Crystals: 20 Yearsof Phonon Transport StudiesUsing Silicon-on-InsulatorTechnologySilicon-on-insulator (SOI) technology has sparked advances in semiconductor andMEMs manufacturing and revolutionized our ability to study phonon transport phenom-ena by providing single-crystal silicon layers with thickness down to a few tens of nano-meters. These nearly perfect crystalline silicon layers are an ideal platform for studyingballistic phonon transport and the coupling of boundary scattering with other mecha-nisms, including impurities and periodic pores. Early studies showed clear evidence ofthe size effect on thermal conduction due to phonon boundary scattering in films down to20 nm thick and provided the first compelling room temperature evidence for the Casimirlimit at room temperature. More recent studies on ultrathin films and periodically porousthin films are exploring the possibility of phonon dispersion modifications in confinedgeometries and porous films. [DOI: 10.1115/1.4023577]

Keywords: silicon, phonon transport, thermal conductivity, nanoscale, thin films

1 Introduction

A major research and manufacturing achievement over the lastdecades of the 20th century was the development and implemen-tation of silicon-on-insulator (SOI) technology, which offersnearly perfect single crystal silicon layers of submicrometer thick-ness attached to a silicon dioxide passive layer on a bulk siliconsubstrate. The resulting SOI wafers have become the starting pointfor a broad variety of innovative devices, including high-performance microprocessors, microfabricated sensors and actua-tors, and photonic crystals [1–3]. The impact has been particularlyprofound for the MEMS community, for which a large fraction ofthe most important device technologies are based on SOI wafersfabricated using either bond and etch back methods or implanta-tion. The silicon dioxide layer between the silicon overlayer andthe bulk substrate serves as an etch stop for fabrication proc-esses—enabling freestanding silicon films in some cases—and asa mechanism for controlling the electric field distribution in nano-transistors. This “buried oxide” also dramatically alters the shapeand magnitude of heat flux lines leaving active regions of thedevices in which it remains, e.g., within SOI nanotransistors foradvanced microprocessors [4].

For the community of researchers interested in micro- andnanoscale solid-state heat conduction, the availability of SOIoverlayers proved to be a watershed event for fundamental pho-non transport research. Prior to the use of SOI substrates in ther-mal transport studies, there was considerable debate over thespecific physics responsible for the reduced conductivity in thindielectric films. The debate was spurred by several reports of verylow thermal conductivity values for non-SOI samples, which wererelatively defective compared to bulk samples. The silicon over-layer provided a nearly defect-free silicon layer for suspensionand characterization, which provides a unique opportunity for per-forming fundamental phonon transport research without the com-plications of process-dependent imperfection scattering. Silicon-on-insulator films have, therefore, become a pivotal platform for

probing phonon-boundary scattering through measurements,many of which determine the effective in-plane thermal conduc-tivity of the silicon layer. Silicon thin films and nanowires of vary-ing dimensions have been studied in depth over the past 20 years[5,6]. In the thinnest films that have been examined experimen-tally (20 nm [7]), boundary scattering reduces the thermal conduc-tivity to as little as �15% of the value for bulk silicon. Thisclassical size effect occurs when the dimensions of the sample aresmaller than the intrinsic mean free path of the material. Theextreme situation when boundary scattering dominates over intrin-sic scattering is often called the “Casimir limit” owing to his earlywork on cylindrical rods [8].

More recently thermal conductivity studies of silicon films pat-terned with periodic arrays of holes provide insight into theimpact of more complex geometries on phonon scattering andpotential modifications the phonon band structure, including thepossibility of phononic crystal effects (which can induce phononband gaps). Nanofabricated thermal sensors and actuators such asradiation detectors (bolometers) [9,10] that rely on thermal insula-tion of particular components to increase the sensitivity, respon-sivity, and accuracy could benefit from the low thermalconductivity of these porous films. These films may also proveuseful for thermoelectric conversion involving the controlledmodification of phonon and electron transport capabilities.

This article reviews experimental measurements of thermal con-duction in thin silicon films, nanowires, and porous films fabricatedfrom SOI wafers. An overview of SOI-based thermal conductivitymeasurement structures is presented in Sec. 2. Section 3 introducesthe thermal conductivity integral model, whose application at themicroscale was first confirmed using measurements on SOI sam-ples. Sections 4, 5, and 6 discuss the thermal conductivity of thinsilicon films, nanobeams, and porous films, respectively.

2 Thermal Conductivity Measurements

Several strategies taking advantage of the SOI structure havebeen developed to measure the thermal conductivity of the singlecrystal device layer. Tables 1 and 2 summarize measurements ofSOI-based thin films and nanowires, respectively.

Manuscript received October 14, 2012; final manuscript received December 20,2012; published online May 16, 2013. Assoc. Editor: Leslie Phinney.

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Table 1 Films (solid and periodically porous) thermal conductivity measurements

Article YearMeasurementconfiguration Doping

Temp.range

Si thickness(nm)

Pore diameter(nm)

Porespacing (nm)

Ju [16] 2005 On-substrate (steady-state jouleheating (1); varying heaterwidth)

p-type Room temp. 20 to 50 — —

Liu andAsheghi [7]

2004 Suspended heater bridge(steady-state)

Undoped 20 K to 300 K 20 and 100 — —

Yu et al. [30] 2010 Suspended heater-thermometer Boron-doping(2� 1019 cm�3)

80 K to 320 K 20 to 25 11, 16, or 270 34 or 385

Liu andAsheghi [25]

2005 Suspended heater bridge(steady-state)

Undoped,2.3� 1020 P,

1.6� 1021 B, or2.3� 1020 As

300 K to450 K

30 — —

Liu et al. [26] 2006 Suspended heater bridge(steady-state)

Undoped 300 K to450 K

50 — —

Hao et al. [18] 2006 Suspended (steady-state jouleheating (1))

p-type(1� 1015 cm�3)

Room temp. 50 to 80 — —

Ju andGoodson [17]

1999 On-substrate (3 -x technique) p-type Room temp. 74 to 240 — —

Aubain andBandaru [15]

2010 Scanning thermoreflectancewith electrical heating

p-type Room temp. 68 to 258 — —

Aubain andBandaru [13]

2011 Scanning thermoreflectancewith electrical heating

p-type(1–10 X cm)

Room temp. 68, 151,235, and 258

— —

Tang et al.[31]

2010 Suspended heater-thermometer Intrinsic,(3� 1014 cm�3 P), or

boron-doped(5� 1019 cm�3 B)

25 K to 300 K 100 32, 81, or 198 55, 140,or 350

Aubain andBandaru [14]

2010 Scanning thermoreflectancewith electrical heating

p-type(14–22 X cm)

Room temp. 260 — —

Asheghi et al.[12]

1998 On-substrate (steady-state jouleheating (3))

< 1015 borondoping

15 K to 300 K 420, 720,and 1420

— —

Asheghi et al.[11]

1997 On-substrate (steady-state jouleheating (3))

n-silicon 20 K to 350 K 420, 830,and 1600

— —

Kim et al. [20] 2012 Suspended (steady-state jouleheating (2))

p-type(boron, 1016 cm�3)

Room temp. 500 �200 to �500 500 to 900

Asheghi et al.[19]

2002 Suspended film (steady-statejoule heating (3))

Boron orphosphorous(1� 1017–3� 1019 cm�3)

15 K to 300 K 3000 — —

Song andChen [21]

2004 Suspended film (steady-statejoule heating (2))

n-type(5� 1014–5� 1015 cm�3)

50 K to 300 K 4670 2300 or 10,900 4000 or20,000

Sverdrup et al.[22]

2001 Suspended film (steady-statejoule heating (4))

— 100 K to200 K

5000 — —

Hopkins et al.[42]

2010 Thermoreflectance(out-of-plane)

(37.5–62.5 X cm) Room temp. 500 300 or 400 500, 600,700, or 800

Table 2 Nanobeam thermal conductivity measurements

Article Year Measurement configuration Temp. range Si thickness (nm) Beam width (nm)

Yu et al. [30] 2010 Suspended heater-thermometer 90 K to 310 K 20 28Bourgeois et al. [27] 2007 Suspended heater bridge

(3 -x)< 2 K 130 200

Heron et al. [28] 2009 Suspended heater bridge(3 -x)

0.3 K to 6 K 100 200

Heron et al. [29] 2010 Suspended heater bridge(3 -x)

< 5 K 100 200 (straightand serpentine)

Hippalgaonkar et al. [32] 2010 Suspended heater-thermometer(array of nanowires)

20 K to 300 K 20 to 100 40 to 150

Boukai et al. [33] 2008 Suspended heater-thermometer(array of nanowires)

100 K to 300 K 2035

10 and 20520

Marconnet et al. [50] 2012 Suspended heater bridge(steady-state)

300 K 196 550

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Early measurements of SOI wafers utilized on-substrate steady-state electrothermal measurement configurations [11,12]. Thesemeasurements utilized three resistive elements as sketched inFig. 1(a). Heat generated at the center element through joule heat-ing is conducted laterally through the silicon layer yielding a non-linear temperature profile symmetric about the center heater line.The thermal healing length, i.e., the characteristic length scale oflateral diffusion within the silicon layer, can be estimated from[11,12]

LH ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidsdoks=ko

p(1)

where do and ds are the oxide and silicon thicknesses, ko is the verti-cal thermal conductivity of the oxide layer, and ks is the in planethermal conductivity of the silicon film. This approximationrequires that the lateral thermal conduction in the oxide layer issmall compared to the lateral conduction in the silicon layer(doko � dsks), the vertical thermal resistance through the siliconlayer is much smaller than that through the oxide layer( ds=ksð Þ � dokoð Þ), and the silicon substrate below the oxide layeris nearly isothermal. The two remaining resistive elements were usedas resistive temperature sensors, which allow measurement of the in-plane silicon thermal conductivity [11,12]. More recently, Aubainand Bandaru [13–15] combined a similar on-substrate joule heatingwith scanning thermoreflectance in order to measure the in-planethermal conductivity. An electrical heater line patterned on the SOIwafers was heated using a sinusoidal current. The amplitude of thetemperature rise in the lateral direction was characterized using scan-ning thermal reflectance and the thermal conductivity extracted usinga time-dependent COMSOL finite element model [13–15].

Additional on-substrate measurement techniques have beendeveloped using varying heater widths and taking advantage of lat-eral spreading within the silicon layer to probe the in-plane conduc-tivity using both steady-state joule heating [16] and harmonic (3x)[17] methods. When the heater width is comparable with or smallerthan the thermal healing length, the temperature rise in the heater isdependent on the in-plane silicon thermal conductivity, while meas-

urements with larger heater widths allow extraction of the contribu-tion of the buried oxide layer.

Suspended structures provide a convenient platform for meas-uring the in-plane conductivity as the heat flow is confined to thelateral direction. Although the required fabrication is more com-plex than for on-substrate measurements, the buried oxide layerprovides a suitable etch stop and/or sacrificial layer for suspendingthe silicon device layer. Several suspended measurement struc-tures have been developed to measure the thermal conductivityand are illustrated in Figs. 1(b)–1(d).

Several groups have used a suspended measurement structure withseveral electrical resistive elements to generate and characterize thetemperature profile in the suspended thin film (see Fig. 1(b)). Theresistive element at the center of the suspended film generates a heatflux that is conducted through the thin film. The temperature profilealong the length of the film is linear and symmetric about the heaterline (as sketched in the bottom panel of the figure). The thermal con-ductivity is estimated from the measured lateral temperature gradientin the film, the applied heating power, and the geometry. To allow forthe assumption of one-dimensional heat conduction from the heaterline, the voltage probes are often spaced near the center of the heaterelement as indicated in Fig. 1(b). Note that if the oxide layer is notremoved during fabrication (e.g., the film is suspended through a back-side etch), the impact of thermal conduction through the oxide layermust be accounted for in the calculation of the thin silicon film con-ductivity. Although estimates of the thermal conductivity are possiblefrom a single heater (e.g., Ref. [18]), more accurate measurements uti-lize one to three additional temperature sensors to characterize thetemperature profile within the thin film [11,12,19–22]. It is importantto note that the measured temperature rise at the heater line may beimpacted by ballistic phonon emission [23] if the heater size is on theorder of the dominant phonon mean free path. For example, Sverdrupet al. [22] observed evidence of these ballistic effects with0.3lm� 3lm heaters at 100 K to 200 K, where the dominant phononmean free path in bulk silicon ranges from 2lm to 10lm, and foundthat the measured temperature rise at the heater line exceeded thatexpected from diffusive transport by up to 60%.

Fig. 1 SOI thermal measurement structures. (a) On-substrate steady-state joule heating structure. (b) Suspended steady-statejoule heating structure. (c) Suspended heater bridge structure. (d) Suspended heater-thermometer structure.

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Suspended heater bridge measurement structures (Fig. 1(c))allow determination of the in-plane conductivity for thin films[7,24–26] and nanowires [27–29]. Specifically the suspended sili-con is coated with a thin metal film that serves as a heater andthermometer during the measurements. The temperature profilealong the length of the suspended film depends on the thermalconductivity of the silicon layer and can be calculated from theheat diffusion equation. The electrical resistance of the metal filmdepends on the temperature profile, and thus, the thermal conduc-tivity of the silicon layer can be extracted from measurements ofthe electrical resistance as a function of applied current. However,care must be taken to account for thermal conduction through themetal film as well. Both steady-state and harmonic (3x) measure-ments are possible with this measurement structure.

Suspended heater-thermometer structures (Fig. 1(d)) have pro-ven useful for characterizing a variety of nanostructures, includingsilicon-based films [30,31] and nanowires [32,33]. The thin filmsor nanowires are suspended between two pads with lithographi-cally patterned resistive heater-thermometer elements. Heat gen-erated at one resistive element conducts through the nanostructureto the second resistive element. The thermal conductivity of thenanostructure is measured using the applied power and the meas-ured temperature difference between the two resistive elements.Care must be taken in the design to ensure that heat generated isprimarily conducted through the nanostructure and not lostthrough the support structure. Thermal resistance between theresistive elements and the nanostructure may impact the thermalconductivity measurements. However, when the resistive elementpads are continuous with the fabricated nanostructure (i.e., bothare patterned from the device layer), the impact of boundary re-sistance should be small. However, when the nanostructure isplaced upon the resistive elements (e.g., Ref. [31]), the thermalboundary resistance leads to a larger temperature differencebetween the resistive elements than would be due to the thermalconductivity of the nanostructure, and thus, the thermal conduct-ance of the nanostructure can be underestimated.

3 Modeling the Thermal Conductivity of Silicon

Much progress has been made in recent years in subcontinuumsimulations of thermal conduction in silicon, including solutionsto the Boltzmann transport equation, lattice dynamics methods,and atomistic calculations accounting for size effects and otherphysical phenomena (e.g., Refs. [34–38]). While experimentalprogress has accelerated in parallel with this work, much of theexisting data (including those obtained using SOI) can be pre-dicted from the semiclassical approach [11,39–41]. This sectionintroduces enough theory to help with the discussion of experi-mental data in subsequent sections. Starting from kinetic theory, asimple model of the thermal conductivity of silicon considers theheat capacity C, the average acoustic phonon velocity vavg, andthe phonon mean free path K, or phonon relaxation time s:

k ¼ 1

3CvavgK ¼

1

3Cv2

avgs (2)

For thin films or nanostructures, boundary scattering reduces thethermal conductivity compared to bulk silicon and a reduced pho-non mean free path and relaxation time can be calculated.

While this simple model in Eq. (2) lumps together the contribu-tion of all phonons, the thermal conductivity integral includes theimpact of the phonon dispersion relations, as well as mode- andwavevector-dependent phonon relaxation times. These thermalconductivity integral models have been used for decades to modelusing the thermal conductivity of silicon, including work on nano-scale thin films [24]. Specifically, the thermal conductivity in thedirection given by the unit vector i can be computed from the ther-mal conductivity integral model [41]:

ki ¼1

2pð Þ3X

j

ðCj q; Tð Þ vj qð Þ � i

� �2sj q; Tð Þdq (3)

where Cj q; Tð Þ, vj qð Þ, and sj q;Tð Þ are the specific heat per phononmode, velocity, and relaxation time of the phonons in branch jwith wavevector q at temperature T.

The spectral-dependent scattering times (mean free paths) forbulk silicon are well documented in literature (e.g., Holland [41]).Typical models for scattering times bulk silicon consider theimpact of Umklapp, impurity, and boundary scattering using Mat-thiessen’s rule. Even for models of bulk silicon, the impact ofboundary scattering must be included at low temperatures wherethe mean free path (due to Umklapp and impurity scattering) isquite long. Neglecting boundary scattering would lead to increas-ing thermal conductivity with decreasing temperature at low tem-perature (<�100 K) where decreasing thermal conductivity withdecreasing temperature is experimentally observed for bulk andnanostructured silicon. As the sample dimensions shrink, bound-ary scattering becomes increasingly important at higher tempera-tures. For example, at room temperature, molecular dynamicsmodels predict that 35% of the thermal conductivity is attributedto phonons with mean free paths longer than 1 lm [35]. Methodsfor accounting for phonon-boundary scattering are discussed inSec. 4 (thin films), 5 (nanobeams), and 6 (porous films).

4 Thin Film Thermal Conductivity

4.1 Impact of Film Thickness. SOI-based metrology techni-ques have allowed the measurement of the thermal conductivityof single-crystal silicon layers from 20 nm to 5 lm thick. Figure 2shows the measured room temperature in-plane and out-of-planethermal conductivity of thin films from the sources listed inTable 1.1 For the thickest films, the in-plane conductivity nears thatof bulk silicon and the thermal conductivity decreases appreciablyfrom the bulk value as the film thickness decreases below �5 lm.Significant variations in the measured results between samples areevident and suggest that other factors, including surface roughness,doping concentration, and film quality, impact the results.

To predict the thermal conductivity reduction due to phonon-boundary scattering, the phonon-boundary scattering relaxation timesj;boundary is estimated from the film thickness and phonon velocity:

s�1j;boundary ¼

vj qð ÞCds

(4)

where C is a correction factor, which must be included to accountfor geometrical factors and the specularity of the surface(e.g., Ref. [41]). This parameter is typically determined from fittingthe models to experimental results. Matthiessen’s rule is then usedto calculate the total relaxation time (s�1

j ¼ s�1j;bulk þ s�1

j;boundary).However, boundary scattering is a surface phenomenon, not an

intrinsic scattering process, so accounting for boundary scatteringusing the above method is not rigorous. Sondheimer [43] derivedfrom the Boltzmann transport equation an analytical expression forcalculating the reduction in electrical conductivity due to boundaryscattering. This method has been adapted to calculating the reducedthermal conductivity within thin films and wires. In the simplestiteration, neglecting spectral variations in the phonon mean freepath and velocity, the reduced thermal conductivity of a thin filmcan be estimated from the conductivity reduction function F:

kfilm

kbulk

� F d; pð Þ ¼ 1� 3 1� pð Þ2d

ð11

1

n3� 1

n5

� �1� exp �dnð Þ

1� p exp �dnð Þ dn

(5)

where d ¼ ds=K is the reduced layer thickness and K is the meanfree path within the film, p is the fraction of phonons specularlyreflected by the boundaries, and n is a variable of integration. Theresults of this model are shown in comparison to the data in Fig. 2for two cases: K¼ 100 nm and K¼ 300 nm. Although simple

1References cited in Table 1 are [7,11–22,25,26,30,31,42].

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estimates of the phonon mean free path from the kinetic theory (Eq.(2)) suggest a mean free path on the order of 40 nm, several studieshave shown that phonons with longer mean free path contribute sig-nificantly to the thermal conductivity. In fact, the Sondheimermodel assuming a longer mean free path (300 nm) provides a betterestimate of experimental data (as shown in Fig. 2).

While simple models consider only a single mean free path forall phonons, the mean free path varies significantly across the pho-non spectrum. The Sondheimer approach can be combined withthe thermal conductivity integral model (Eq. (3)) in order to con-sider the spectral dependence of phonon-boundary scattering[11,12,19]. In this case, the bulk scattering time for each modesj;bulk q;Tð Þ is reduced by the conductivity reduction functionF d; pð Þ:

sj q; Tð Þ ¼ F d; pð Þsj;bulk q; Tð Þ ¼ Fds

Kj;bulk q;Tð Þ ; p� �

sj;bulk q; Tð Þ

(6)

Note that the reduction function F is that of Eq. (5) (for thinfilms), where the reduced thickness d is mode dependent. Amode-dependent specularity parameter can also be determinedfrom the surface roughness and the wavelength of each phononmode [44] or a single value of specularity can be assumed for allmodes (e.g., p¼ 0 for purely diffuse scattering).

4.2 Impact of Temperature. Figure 3 shows the temperaturedependence of the thermal conductivity of silicon thin films(dS¼ 20 nm, 420 nm, and 1.42 lm [7,12]) in comparison to themeasured value for bulk silicon [45]. The results of a thermal con-ductivity integral model considering the spectral-dependent dif-fuse boundary scattering (Eqs. (3), (5), and (6)) shows goodagreement with the measured values.

At low temperatures, the phonon mean free path is limited byboundary scattering and the thermal conductivity follows the de-pendence of the heat capacity. At these temperatures, much lowerthan the Debye temperature, the phonon heat capacity follows aT3 dependence as the phonon population increases with increasingtemperature. The impact of boundary scattering is more pro-nounced at low temperatures than at room temperature becausethe intrinsic mean free path is much longer.

As the temperature increases, the phonon population continues toincrease, but phonon scattering rate also increases. A peak in thethermal conductivity is observed as the increased scattering begins to

outweigh the impact of increasing heat capacity. The peak thermalconductivity shifts to higher temperatures with decreasing film thick-ness. At high temperatures, the thermal conductivity decreases withincreasing temperature as the phonon mean free path decreases.

Fig. 3 Temperature-dependent thermal conductivity of severaldifferent SOI-based silicon structures: thin films (Asheghi andcolleagues [7,12]), 20 nm 3 28 nm rectangular nanobeams (Yuet al. [30]), and 22 nm thick nanoporous films (results shown forboth 11 and 16 nm diameter holes spaced by 34 nm from Yuet al. [30]). The thermal conductivity of bulk silicon (Ho et al.[45]) is shown for comparison. While the modeling results agreefairly well for the thin film data, the nanobeam and nanomeshresults fall below the predicted thermal conductivities.

Fig. 2 Thickness dependence of the thermal conductivity ofsilicon thin films [7,11–22,25,26,30,31,42]. For the reported in-plane thermal conductivity data; red rings around the solid cir-cular data markers indicate nearly pure samples (intrinsic,nearly pure, or < 1015 cm23 dopant atoms). The Sondheimermodel (Eq. (5)) for the reduced thermal conductivity as a func-tion of film thickness is shown for a mean free path of 100 nmand 300 nm, assuming purely diffuse scattering (p 5 0) at thefilm boundaries.

Fig. 4 Impact of doping on the thermal conductivity of (a) 3 lmand (b) 30 nm thick silicon films. Figures reprinted with permis-sion from (a) Asheghi et al. [19] and (b) Asheghi and Liu [25].

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4.3 Impact of Doping. Doping of single-crystal silicon filmsis required for many applications, including semiconductor devi-ces. Both intentional and unintentional impurities cause increasedphonon scattering and reduce the thermal conductivity. Thus,comparing measured thermal conductivities from different authorsis complicated by the range of doping levels and dopant atomsconsidered (see Table 1). A few authors have studied the impactof doping by measuring samples with different doping species andconcentrations. Figure 4 illustrates the impact of doping for 3 lmthick films doped with various concentrations (1� 1017 to3� 1019 cm�3) of boron or phosphorous [19] and 30 nm filmsdoped with boron (1.6�m1021 cm�3), arsenic (2.3� 1020 cm�3),or phosphorous (2.3� 1020 cm�3) from 300 K to 450 K [25].

4.4 Beyond the Classical Size Effect. Recent work on ultra-thin films has shown that the dispersion relations can be modifiedby the confined geometry. Raman scattering spectra for suspendedsilicon membranes (24 nm to 32 nm thick) have shown evidenceof confined phonon modes [46]. Additionally, in silicon mem-branes (8 nm to 32 nm thick), angle-resolved Brillouin scatteringspectroscopy measurements show a strong reduction in the pho-non velocities for the fundamental flexural mode compared tobulk silicon [47]. In addition to the boundary scattering effects onthe phonon mean free paths, these types of modifications to thedispersion relation can impact the thermal conductivity of nano-scale thin film and other confined nanostructures.

Ballistic effects are also important to consider when the dimen-sions of the heat source are comparable to the phonon mean freepaths. In SOI-based measurements, nondiffusive thermal transportin silicon films has been observed using electrothermal [22] andthermal grating [48,49] techniques and for bulk silicon using ther-moreflectance measurements [49]. These measurements provideinsight into the distribution of mean free paths and their contribu-tion to thermal conductivity. Both these experimental measure-ments and models (e.g., Ref. [35]) have shown that phonons withvery long mean free paths (K > 1lm) contribute significantly tothe thermal conductivity.

5 Nanobeam Thermal Conductivities

Rectangular cross-section nanobeams can be etched from thesilicon device layers and directly integrated into the suspendedheater-thermometer measurement structures or the suspendedheater bridge configuration, both of which are described in Sec. 2.In the heater-thermometer configuration, SOI-based manufactur-

ing (with the nanowire directly integrated with the heating\sensingpads during fabrication) provides an advantage over measure-ments of other nanowires that must be placed onto the resistiveelements. In the latter case, thermal boundary resistance betweenthe nanowire and the resistive element can lead to errors inextracting the thermal conductivity and this effect is minimized inthe SOI-based structure. Table 2 summarizes experimental meas-urements of silicon nanobeams.2

The phonon mean free path due to boundary scattering in theCasimir limit [8] is the nanowire diameter Dw for circular nano-wires and 1.12*W for square nanowires with a side width W. Thecombined impact of boundary scattering and intrinsic scatteringcan be evaluated using Matthiessen’s rule and the relative reduc-tion to the thermal conductivity can be approximated as [51]

knanowire

kbulk

� 1þ KDw

� ��1

(7)

for circular nanowires. For rectangular nanobeams of width W andthickness ds, a critical thickness (or effective diameter) can bedefined [32] as dc ¼ 2

ffiffiffiffiffiffiffiffiffidsWp

=ffiffiffipp

(which reduces to dc � 1:12Wfor square nanobeams). This dimension can replace the nanowirediameter in Eq. (7) to estimate the reduced thermal conductivityof noncircular nanobeams. However, as shown in Fig. 5(a), theexperimentally measured room temperature thermal conductivityof several rectangular nanowires does not appear to follow thetrend of Eq. (7). Although there are few experimental data points,the thermal conductivity appears to scale with d2

c , which meansthat the thermal conductivity scales proportional to the nanowirecross-sectional area. For comparison, the thermal conductivity ofrough [52] and smooth [53] cylindrical nanowires are also shownin Fig. 5(a). The reduced thermal conductivity of the rough nano-wires demonstrates the impact of surface quality on the thermalconduction in nanostructures.

Similar to the thin films described above, for purely diffusescattering in a nanowire with an arbitrary cross section Ac, theconductivity reduction function can be calculated [43]:

F K;p¼ 0ð Þ¼ 1� 3

4pAc

ðAc

ð2p

0

ðp

0

sinhcos2 hexp�LOP

K

� �dhd/dAc

(8)

Fig. 5 Thermal conductivity of silicon nanobeams [30,32,33] as a function of (a) critical thick-ness and (b) temperature. The thermal conductivity of rough [52] and smooth [53] cylindricalnanowires are shown in panel (a) for comparison to the nanobeam data. The results of the sim-ple model for nanowire thermal conductivity from Eq. (7) are shown with the solid line in panel(a), while the data for the rectangular nanobeams appear to follow an approximate trend ofk � d2

c . In panel (b), the temperature-dependent thermal conductivity results from a thermalconductivity integral model with the Sondheimer-type reduction function to account for theboundary scattering in rectangular nanobeams are shown for in comparison to the experimen-tal data. The large nanowires from Boukai et al. [33] fall significantly higher than the model fornanobeams (and also the prediction for 35 nm thick films), while the smaller nanowires fromBoukai et al. [33] and Yu et al. [30] fall below the predictions.

2References cited in Table 2 are [27–30,32,33,50].

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where LOP is the distance from a point O on the cross section ofthe nanowire to a point P on the surface of the nanowire along thedirection given by the azimuthal angle h and radial angle /. Con-ductivity reduction functions have been derived from Eq. (8) forcircular [43] and rectangular nanowires [40,54]. For circular nano-wires, the approximate solution from the Casimir limit and Mat-thiessen’s rule (Eq. (7)) agrees well with the conductivityreduction function (Eq. (8)) [55]. The spectral dependence of theconductivity can be taken into account using the thermal conduc-tivity integral approach (Eqs. (3) and (6)).

The temperature-dependent thermal conductivities of severalrectangular nanowires are shown in Fig. 5(b). The results of athermal conductivity integral model (using a Sondheimer-typereduction function) are shown in comparison to the experimentaldata for four nanobeam cross sections. Thermal conductivity inte-gral models reducing the bulk mean free path with Matthiessen’srule and the critical thickness do not vary significantly in magni-tude or trend from these results. While the model agrees well withthe magnitude and trend of the data from Hippalgaonkar et al.[32], the remaining data fall far from the predicted values. Thethermal conductivities of the larger nanowire from Boukai et al.[33] are underpredicted by the models, and the experimental dataare actually larger than those of thin films of the same thickness[7]. In contrast, the measured thermal conductivities of the smallernanowires [30,33] are overpredicted by the models.

Adding serpentine kinks to nanobeam geometry has beenshown to reduce the thermal conductance by up to 40% at verylow temperatures (T< 5 K) [28,29]. A spectral-dependent modelfor the thermal conductance suggests that the origin of this reduc-tion is that the serpentine structures block a small fraction of thephonons that would have had very long mean free paths in thestraight nanowires [29].

6 Aligned Porous Thin Film Thermal Conductivity

Silicon films with controlled, periodic arrays of cylindricalholes can be fabricated by patterning the SOI device layer. Thesecontrolled, porous films have proven useful for nanophotonicdevices [56–59] and are promising for a number of other emergingtechnologies [60], including high ZT thermoelectric materials[30,31,61]. Measurements of the aligned porous thin films aresummarized in Table 1. The introduction of the pore structure cansignificantly reduce the thermal conductivity of the silicon layercompared to solid films of the same thickness (see Fig. 6). Thetemperature dependence of the thermal conductivity for a nanopo-rous film from Yu et al. [30] is included in Fig. 3.

A key challenge in measuring the thermal conductivity of theporous silicon arises from the highly nonuniform cross-sectionalarea for heat transfer. Many measurements actually measure the

thermal conductance across the nanoporous film. Several correc-tion methods for estimating the solid thermal conductivity fromthe measured value (typically conductance) have been developed,including using the Eucken factor [62], corrections based on finiteelement analyses [20], normalizing the data by the porosity [31],or determining a upper bound on thermal conductivity by consid-ering only the minimum cross section for thermal conduction(Amin ¼ S� Dð Þds) [30]. The variety of correction methods usedmakes comparing data sets from different authors challenging. Inthis work, the solid thermal conductivities as reported by theauthors are plotted.

Two-dimensional periodically porous films have been studiedacross a range of dimensions (film thickness ds, hole diameter D,and hole spacing S). Early samples fabricated with conventionalphotolithography techniques yielded samples with �2 lm to10 lm diameter holes in �5 lm thick silicon [21]. In more recentmeasurements, hole diameters as small as 11 nm have beenachieved through a superlattice nanowire pattern transfer (SNAP)technique [30]. Despite advances in fabrication techniques overthe last decade, the largest achievable aspect ratio for the holesappears to be limited to ds=D < 3.

In two-dimensional porous films, key dimensions include thefilm thickness, hole diameter, and hole spacing. The introduction ofnano- or microscale pore structures leads to a large increase in thedensity of interfaces within the structure and increased phonon scat-tering. As with the nanowires, the impact of geometry on phonontransport can either be included by reducing the mean free pathwith Matthiessen’s rule or through a conductivity reduction func-tion. For microporous and nanoporous solids, Hopkins et al. [62,63]utilize a phonon-pore boundary scattering term based on the limit-ing dimension in the systems (spores ¼ Ld=vj qð Þ), where Ld is thelimiting dimension. For thermal transport, the intrapore distance(pore edge-to-pore-edge distance, or “neck width”) is often consid-ered the limiting dimension Ld ¼ S� D in the system as it repre-sents the minimum cross section for heat transport. Figure 7(a)shows the thermal conductivity as a function of limiting dimensionfor several porous films. Also shown is the recent measurement[50] of periodically porous nanobeams, which are a 1D analog tothe 2D films. Because of the additional side wall boundaries fornanobeams, the limiting dimension is the minimum of the intraporedistance and the distance from the pore edge to the edge of thenanobeam. A clear decrease in the thermal conductivity is observedas the limiting dimension is reduced. The results of a thermal con-ductivity integral model considering the limiting dimension usingMatthiessen’s rule is shown in Fig. 7(a). As the limiting dimensiondepends only on the pore geometry, the results of this model areindependent of film thickness. As the small limiting dimensionsamples were generally fabricated from thinner films, the overesti-mation of the thermal conductivity at small limiting dimensionscould be associated with increased scattering due to the film boun-daries. Furthermore, the limiting dimension model does not takeinto account the varying cross section for heat transport and otherdetails of the geometry. However, this highly spatially nonuniformgeometry does not easily lend itself to calculations of aSondheimer-type thermal conductivity reduction function analo-gous to Eq. (5) for solid thin films. Monte Carlo-type methods forestimating the reduced phonon mean free path are proving useful tocalculating the reduced thermal conductivity in nanostructures andhave been applied to porous nanobeams [50].

In classical effective medium theories for porous solids (e.g.,Eucken), the thermal conductivity depends only on the porosity ofthe film. Figure 7(b) shows the thermal conductivity of two-dimensional periodic porous films as a function of porosity. Earlywork by Song and Chen [21] showed that, contrary to the classicalmodels, as the temperature decreased the ratio of kporous=kbulk

decreased and hypothesized that the increasing phonon mean freepath with decreasing temperature led to the deviation. Morerecently, Tang et al. [31] showed that for constant porosity (35%),the thermal conductivity decreases with decreasing hole size. Fur-thermore, comparing the results of Yu et al. [30] with dimension

Fig. 6 Room temperature thermal conductivity of 2D periodi-cally porous thin films [20,21,30,31] and 1D periodically porousnanobeams [50] as a function of the film thickness. The porousfilm data are compared to the predictions from Eq. (5) forin-plane thermal conductivity of solid films.

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on the order of 10 nm to that of Kim et al. [20] with dimensionson the order of 100 nm, a clear size effect is evident beyond theimpact of porosity. These results highlight the need to use and de-velop detailed models of thermal conduction in these nanoporoussystems that take into account the impact of varying mean freepaths and scattering size effects.

Furthermore, several groups [30,42,61,63] have hypothesizedthat these periodic, porous structures impact the phonon bandstructure leading to an additional reduction in the thermal conduc-tivity beyond that due to boundary scattering. There have beenextensive theoretical studies of the phonon dispersion modifica-tions due to periodic pore structures [60,63–66]. In particular,periodically porous films have been shown to lead to phononbandgaps (e.g., Ref. [67]) and these structures are sometimesreferred to as “phononic crystals.” Even in the absence of pho-nonic bandgaps, the porous structure has been predicted to modifythe phonon dispersion leading to reduced phonon group velocityand density of states (e.g., Ref. [68]). However, because of the dif-ficulties in accurately modeling the phonon-pore scattering (evenfor idealized pores), it has not been proven unequivocally that thecoherent effects are responsible for any additional reduction inthermal conductivity of these porous, thin films.

7 Concluding Remarks

Twenty years ago, there was a surging interest in microscale andnanoscale heat conduction physics, including a variety of manu-scripts in the Journal of Heat Transfer and elsewhere on phonontransport. There was, however, a troubling lack of experimentaldata or, when data were available, it was challenging to determinewhether the governing physics were related to processing details orto basic physics. The use of SOI technology—while being inargu-ably more important for MEMS development—fundamentallychanged this situation by providing nearly defect-free, high quality,thin silicon films. Measurements of SOI-based silicon films span-ning three orders of magnitude in thickness have shown clear evi-dence of the size effect matching well with semiclassical modelsfor thermal conductivity, including phonon-boundary scattering.Over the past two decades, SOI technology, combined withimproved lithography techniques, have provided a robust platformfor studying the complex phonon transport in geometries includingstraight and bent nanobeams and periodically porous films.

Nomenclature

Ac ¼ cross-sectional area (m2)C ¼ heat capacity (J m�3 K�1)

Cj ¼ heat capacity per phonon mode (J m�3 K�1)D ¼ hole diameter (m)

Dw ¼ nanowire diameter (m)DH ¼ hydraulic diameter (m)dc ¼ critical thickness (m)ds ¼ silicon device layer thickness (m)do ¼ buried oxide layer thickness (m)F ¼ conductivity reduction functioni ¼ unit vector in the ith directionk ¼ thermal conductivity (W m�1 K�1)

ks ¼ silicon thermal conductivity (in-plane for films; axialfor nanowires) (W m�1 K�1)

ko ¼ oxide thermal conductivity (W m�1 K�1)Ld ¼ limiting dimension (m)Lh ¼ thermal healing length (m)

LOP ¼ distance from point O on cross section of thenanowire to a point P on the nanowire boundary

p ¼ specularityq ¼ phonon wavevector (m�1)S ¼ hole spacing (m)T ¼ temperature (K)

vavg ¼ average acoustic phonon velocity (m s�1)W ¼ nanobeam width (m)

Greek Symbols

C ¼ correction factor for boundary scatteringd ¼ reduced layer thickness, d ¼ ds=Kh ¼ azimuthal angle (radians)K ¼ phonon mean free path (m)n ¼ integration variables ¼ phonon relaxation time (s)/ ¼ radial angle (radians)x ¼ phonon angular frequency (s�1)

Subscripts

boundary ¼ due to boundary scatteringbulk ¼ due to bulk/intrinsic processesfilm ¼ of the thin film

i ¼ direction indexj ¼ phonon branch index

nanowire ¼ of the nanowirepores ¼ due to phonon-pore boundary scattering

porous ¼ of the porous film

Fig. 7 Room temperature thermal conductivity of 2D periodically porous thin films [20,21,30,31]and 1D periodically porous nanowires [50] as a function of (a) the limiting dimension and (b) theporosity. In panel (a), for the films, the limiting dimension is the intrapore distance (S-D). For the1D porous nanoladders Marconnet et al. [50], the limiting dimension is the smaller of the intra-pore distance and the distance from the edge of the nanowire to the pore wall, (W-D)/2. Film thick-nesses ds are indicated in the legend. The thermal conductivity data are compared to the resultsof the thermal conductivity integral model with the mean free path reduced using Matthiessen’srule and the limiting dimension. The results of the thermal conductivity integral model are inde-pendent of film thickness.

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